13_AS_8_lec_a

G 2 3 because of the constraints imposed by choice of

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Unformatted text preview: Statistical Tests The Effect of Duplicate policies 67/72 Actuarial Statistics – Module 8: Method of Graduation Methods of graduation Statistical Tests Statistical Tests (revisited) I The following six tests can be used to test the adherence of a graduation to data: chi-square test standardised deviations test sign test cumulative deviation test grouping of signs test serial correlations test. Degrees of freedom in the Chi-square test to test the adherence to data of a graduation depends on the method of graduation: lose one degree of freedom for each parameter fitted when using a parametric formula 67/72 Actuarial Statistics – Module 8: Method of Graduation Methods of graduation Statistical Tests Statistical Tests (revisited) II when graduating using a standard table lose one degree of freedom for each parameter fitted also lose some degrees of freedom (e.g. 2 - 3 ) because of the constraints imposed by choice of the standard table for graphical graduation it is difficult to determine the degrees of freedom lost - a rule of thumb is 2 or 3 degrees of freedom for every ten or so ages 68/72 Actuarial Statistics – Module 8: Method of Graduation Methods of graduation Statistical Tests Tests: Comparing an experience with a standard table s Notations: rates available from standard tables are qx or µs + 1 x 2 Null hypothesis H0 : the true mortality rates are same as those in the standard tables All the following six tests can be used to test goodness of fit of a set of estimated rates against a standard table (use the standard table rates to calculate the expected number of deaths): chi-square test, standardised deviations test, sign test, cumulative deviation test, grouping of signs test, serial correlations test. Testing goodness of fit but not under or over graduation For the χ2 -test, the statistic is a χ2 random variable with the degree of freedom equal the no. of age groups. 69/72 Actuarial Statistics – Module 8: Method of Graduation Methods of graduation The Effect of Duplicate policies 1 Introduction 2 Testing smoothness 3 Statistical tests Preliminaries Chi-square (χ2 ) test Standardised Deviations Test Signs test Cumulative Deviations Test Grouping of Signs Test Serial Correlations Test 4 Methods of graduation Preliminaries Graduation by Parametric Formula Graduation by Reference to a Standard Table Graphical graduation Statistical Tests The Effect of Duplicate policies 70/72 Actuarial Statistics – Module 8: Method of Graduation Methods of graduation The Effect of Duplicate policies The Effect of Duplicate policies For insured lives investigations we observe policy years and numbers of policies that are death claims and not the lives This gives rise to duplicate policies and the assumption of independence does not hold Assume N lives from age x to x + 1 Assume proportion πi lives own i insurance policies (these proportions are unknown) Total number of policies is i πi N i 70/72 Assume the mortality rate for each life is qx Let Di be the number of deaths among the πi N lives each with i policies and Ci be the number of claims among these lives Actuarial Statistics – Module 8: Method of Graduation Methods of graduation The Effect of Duplicate policies Assuming the Binomial model, we have Di ∼ Bin (πi N , qx ) We then have E [C ] = E Ci = E iDi i = i iE [Di ] = i πi Nqx i i And var [C ] = var Ci = var i i 2 var [Di ] = = i 71/72 iDi i i 2 πi Nqx (1 − qx ) i Actuarial Statistics – Module 8: Method of Graduation Methods of graduation The Effect of Duplicate policies If all have i i πi N death claims were independent then we would E [C ] = i πi N qx i var [C ] = i πi N qx (1 − qx ) i Duplicate policies increase the variance of the number of claims by the ratio 2 i i πi i i πi 72/72 which differs for each age We could make allowance for the increased variances in statistical tests, if the variance ratios are known The CMIB estimates these ratios: typically between 1.18 to 1.75, average 1.46...
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